Spline-Interpolation

In the spline interpolation trying to given reference points, called nodes to interpolate with piecewise continuous polynomials, splines accurate. While the result of a polynomial interpolation often oscillated by unfavorable fixed reference points beyond recognition, yet useful curves and approximation properties provides the spline interpolation with only a small computational effort even ( Runge phenomenon). The spline interpolation can be calculated with low linear effort, but it provides compared to the polynomial a lower order of convergence.

Template for the spline interpolation ( third degree ) is the traditional, flexible ruler of shipbuilders, the Straklatte (English spline). This is fixed at any number, given by the designer points and then connecting the points by a smooth and harmonious bending line. The Straklatte generated as the line through all the points with minimum bending energy and minimum curvatures. While the Straklatte the turning points ( locations of maximum linearity and minimum bending energy) are generally between the nodes and the nodes themselves, locations of maximum curvature are ( locations of maximum force by fixation), the turning points are at the Polynomeninterpolation close to the nodes in the polynomial best approximation even in the nodes.

The terms spline interpolation and spline function without further additions always denote the spline / spline of third degree. Both terms are usually used interchangeably. The term spline is, however, increasingly as an abbreviation for B -spline, rare for other splineartige lines such as the Bezier curves used.

Simple approach ( polyline )

The simplest method is to use straight line between two adjacent points, the calculation of a simple polygonal line as the splines is effected in the same manner, with the one calculated and the graph between two points:

It is clear that this " simple" polynomial splines - can be very inaccurate - as mentioned above. Much better results cubic spline polynomials.

The cubic C2 spline

Cubic splines are splines that on each subinterval ( ie between two points) correspond to a cubic polynomial. They are twice continuously differentiable () and satisfy a minimality for the second derivative, which makes them particularly interesting to other splines.

For the interpolation of the function calls you now. The cubic splines are suitable due to their smoothness suitable for the approximation of "smooth" functions. Due to their design, they tend not to overshoot unlike interpolation polynomials.

On each subinterval is chosen then the polynomial in Newton representation to set the spline interpolation.

In order to solve the system of equations clear conditions are required. For each of the two intervals interpolation conditions are met:

This creates conditions. Other conditions are obtained by a spline that at all internal nodes must be twice continuously differentiable:

For the remaining two conditions (boundary conditions ) there are various ways, such as:

• Free edge or natural spline.
• Clamped edge :, where and predetermined, usually the same either by the derivative of the function to be interpolated or by an approximation.
• Periodic boundary condition:
• Not-a -knot ( used for example by the Matlab program ): The outer three points each interpolated by a polynomial.

The first derivative (slope) looks like this:

The second derivative ( curvature ) looks like this:

Above obvious approach requires some computational work.

Barycentric coordinates

The use of barycentric coordinates gives better overview. Within the interval are introduced to the auxiliary variables and the barycentric variables. All leads are after, and so. The reference point is the right edge of the interval and at the left margin in case. The results of the new approach

By the continuity is already established.

The first derivatives are

Now is the abbreviation for the unknown parameters. From the connection requirements for, and thus follows

So and.

The second derivatives are

And thus the connection requirements for

And

This leads to the tridiagonal, strictly diagonally dominant linear system of equations

With two free parameters, such as and.

For equidistant sampling points with distance to the system of equations simplifies to

Here, you can extend the equations as follows symmetrically

Which corresponds to the clamping and the tridiagonal Toeplitz matrix

Leads. This has the inverse

With coefficients of the equations, and the recursion explicitly the formula

Suffice. Transmission in dimensions greater than one, for example, rectangular grid, presents no difficulties.

Properties

In 1957, Holladay proved the following named after him identity of Holladay. With the space of the two times differentiable functions is referred to for which the zeroth and first derivative are continuous and the absolute second derivative is. Be an interpolating spline to and the standard, the following applies

With.

Meets the natural spline function, periodic or complete boundary conditions, as is so

Minimality of Splineinterpolierenden: whether and fulfill one of the three boundary conditions, then applies

Interpolation with shape retention

Splines are widely used due to their properties in CAD. This raises the question under which conditions a spline interpolant inherits one of the following shape-preserving properties of the interpolating function:

• Non-negativity: for all
• Monotonicity: for
• Convexity: for all and

This shows that classical splines have something worse properties than Bezier curves. First, there is the question of when an interpolating spline is convex.

For classical splines, that is the set of possible splines on the interval to the grid is a finite vector space. (Not necessarily coincident with the grid ) can be specified and demanded that the spline is continuously differentiable in and beyond for valid node and associated ordinates for the interpolation. One demands in addition the convexity of the interpolating splines and minor technical assumptions, it is found that the set of all Ordinatentupel for which there is such a spline is complete.

This has far- reaching consequences. is a proper subset of the case, because the input data do not need to be in a convex position. Specification of a tuple on the edge of computing as a result of inaccuracies or other defects, the amount having been abandoned, so that despite the solvability of the original problem no solution is found. The other implication of the sentence is even worse. These five points are arranged in the form of the character " " so that the middle point lies exactly on the top. The only convex interpolant is then the sum function, and this is not continuously differentiable. So the 5-tuple belongs to the complement of, and this is open. Thus there is a neighborhood of the 5- tuple, in which it also is not convex, continuously differentiable interpolant. If you move the mid-point slightly upwards, without leaving the environment, then we obtain hence five points in strictly convex position, which yet the interpolation problem has no solution. Since this effect increases in default of many interpolation, remains only one way to ensure the solvability of input data in strictly convex position, that is to violate the conditions of the sentence. The amount from which the splines may be taken, should not be a finite dimensional vector space. For this offer to inter alia:

• ( broken - ) rational splines
• Splines with arbitrary intermediate nodes
• Exponentialsplines
• Lacunar ( incomplete ) splines